91Ó°ÊÓ

The relation between two coordinate systems corresponding to two reference frames that are moving relative to each other can be represented in the form of a matrix known as the Lorentz transformation matrix. It is expressed as,

$\left[\begin{array}{c}{x}^{\text{'}}\\ y^{\text{'}}\\ z^{\text{'}}\\ c{t}^{\text{'}}\end{array}\right]\text{ }=\text{ }\left[\begin{array}{cccc}{\mathrm{γ}}_v& 0& 0& −{\mathrm{γ}}_v\frac{v}{c}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ −{\mathrm{γ}}_v\frac{v}{c}& 0& 0& {\mathrm{γ}}_v\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ ct\end{array}\right]$

The transformation matrix relation for classical limit (i.e.$\frac{v}{c}<<\text{ }1\text{ }⇒\text{ }{\mathrm{γ}}_v\text{ }â‰\dots \text{ }1$,) can be expressed as,

localid="1659092294568" $\left[\begin{array}{c}{x}^{\text{'}}\\ y^{\text{'}}\\ z^{\text{'}}\\ c{t}^{\text{'}}\end{array}\right]=\text{ }\left[\begin{array}{cccc}1& 0& 0& −\frac{v}{c}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ −\frac{v}{c}& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ ct\end{array}\right]$

The resulting equations that we get,

$\begin{array}{c}{x}^{\text{'}}\text{ }=\text{ }x\text{ }−\text{ }vt\\ y^{\text{'}}\text{ }=\text{ }y\\ z^{\text{'}}\text{ }=\text{ }z\\ t^{\text{'}}\text{ }=\text{ }t\end{array}$

The last equation for$t^{\text{'}}$, $\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$c$}\right.$is approximated to zero because of the classical limit. The Lorentz transformation matrix thus, results in the Galilean transformation matrix for the classical limit.